Returning to the case where \(G\)  is a reductive group acting linearly on a projective scheme \(Y\) , if \(\emptyset \ne Y^{s} \subsetneq Y^{ss}\) , then \(C(Y^{ss},G)\)  is a proper closed \(G\) -invariant subscheme of \(Y^{ss}\)  which is disjoint from \(Y^{s}\) . Assuming that \(Y\)  is irreducible for simplicity, the first stepIn fact, this first step in [1]} is performed as a sequence of blow-ups: one chooses representatives of all conjugacy classes of connected reductive subgroups of \(G\)  of dimension \(d_{\max }(Y^{ss},G)\)  and for each such subgroup \(R\)  one blows-up the closure of the \(G\) -sweep of the locus of semistable points which are fixed by \(R\) . in the partial desingularisation procedure in [1]} is to blow-up \(X\)  along the closure of \(C(Y^{ss},G)\)  to obtain \(\varphi _1: \widetilde{Y}_{1} \rightarrow Y\) . The \(G\) -action on \(Y\)  naturally lifts to \(\widetilde{Y}_{1}\)  and by pulling back the ample linearisation \(\mathcal {L}\rightarrow Y\)  and twisting by a small multiple of the exceptional divisor one obtains an ample \(G\) -linearisation \(\mathcal {L}_1 \rightarrow \widetilde{Y}_1\)  such that
\( \varphi _1^{-1}(Y^s) \subset \widetilde{Y}_1^s \subset \widetilde{Y}_1^{ss} \subset \varphi _1^{-1} (Y^{ss})\) 
